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Find the area of a regular hexagon with an apothem 10.4 yards long and a side 12 yards long. Round your answer to the nearest tenth.
(3 points)
To find the area of a regular hexagon, you can use the formula: Area = (3 * √3 * s^2) / 2,
where s is the length of each side of the hexagon.
Given that the side length is 12 yards, and the apothem is 10.4 yards, we need to find the length of the apothem in terms of the side length.
In a regular hexagon, the apothem is equal to the height of an equilateral triangle formed by two consecutive sides of the hexagon.
Using the Pythagorean theorem, we can find the height of this equilateral triangle:
height^2 = side^2 - (side/2)^2
height^2 = 12^2 - (12/2)^2
height^2 = 144 - 36
height^2 = 108
height ≈ 10.4 yards
Now we substitute the side length into the formula for the area of a regular hexagon:
Area = (3 * √3 * 12^2) / 2
Area = (3 * √3 * 144) / 2
Area = 216√3 / 2
Area = 108√3
To find the decimal approximation, we can use a calculator:
Area ≈ 187.771
Therefore, the area of a regular hexagon with an apothem of 10.4 yards long and a side of 12 yards long is approximately 187.8 square yards.